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Solution of Nonlinear Systems of Equations

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Boundary Value Problems for Engineers

Abstract

In previous chapter, we worked on problems related to determine the value of unknown x that satisfied a single scalar equation, \( f\left( x \right) = 0. \)

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Notes

  1. 1.

    A simple approach for solving nonlinear systems of algebraic equations (which is called successive substitution) is to use the same strategy that is employed for fixed-point iteration , where, each one of the nonlinear equations can be solved for one of the unknowns, and they can then be implemented iteratively to compute new values until they converge on the solutions. Shortcomings of successive substitution method are that the convergence often depends on the manner in which the equations are formulated and divergence can occur if the initial guesses are selected far from the actual solution. Therefore, successive substitution iteration has limited utility for solving nonlinear systems and we do not include sample solutions of problems related to this method.

  2. 2.

    We use both names interchangeably for the same method that has been employed for solving single and multi-dimensional nonlinear equations.

  3. 3.

    For example, Broyden’s method replaces the Jacobian matrix in Newton’s method with an approximation matrix that is updated at each iteration , requiring only m scalar functional evaluations per iteration at the expense of the lost quadratic convergence of Newton’s method.

  4. 4.

    Readers may refer to Optimization Toolbox documentation of Mathworks, Inc., for more examples and how to solve Nonlinear Equations with Analytic Jacobian , Nonlinear Equations with Finite-Difference Jacobian , Nonlinear Equations with Jacobian , and Nonlinear Equations with Jacobian Sparsity Pattern. Details of algorithms for Trust-Region Dogleg Method, Trust-Region Reflective fsolve Algorithm, Levenberg-Marquardt Method are also described in the same documentation.

  5. 5.

    See, Mathworks web site for more information. (https://www.mathworks.com/help/optim/ug/when-the-solver-fails.html) last time accessed 15th, March 2019.

  6. 6.

    Detailed presentation of this subject can be found in [7,8,9,10,11,12].

  7. 7.

    This parameter is represented in literature using different symbols, \( t, \lambda \) being the most common ones.

  8. 8.

    Some books use the term “Convex homotopy ”, such as in [16].

References

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Authors and Affiliations

  1. Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey

    Ali Ümit Keskin

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  1. Ali Ümit Keskin
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Correspondence to Ali Ümit Keskin .

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Keskin, A.Ü. (2019). Solution of Nonlinear Systems of Equations. In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_2

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  • DOI: https://doi.org/10.1007/978-3-030-21080-9_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21079-3

  • Online ISBN: 978-3-030-21080-9

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